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Консультация № 200249
09.02.2021, 17:05
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Написать канонические уравнения прямой 5x+y+2z+4=0, x-y-3z+2=0
Написать канонические уравнения прямой 5x+y+2z+4=0, x-y-3z+2=0
Обсуждение
14.02.2021, 07:48
общий
это ответ
Здравствуйте, ALX5!
Каноническое уравнение прямой в пространстве имеет вид
где (x[sub]0[/sub], y[sub]0[/sub], z[sub]0[/sub]) - произвольная точка прямой, {l, m, n} - направляющий вектор прямой. Если прямая является пересечением двух плоскостей, уравнения которых известны (как в данном случае), то в качестве направляющего вектора можно взять векторное произведение нормальных векторов этих плоскостей, которое по определению перпендикулярно обоим векторам, следовательно, параллельно плоскостям, а значит, и прямой. В качестве точки можно взять любую, чьи координаты удовлетворяют уравнениям обеих плоскостей.
Координаты нормальных векторов определим непосредственно из уравнений соответствующих плоскостей:
Их векторное произведение будет равно
то есть направляющий вектор прямой {-1, 17, -6}. Координаты точки получаем, решая систему
Её решением будет, например, x = -1, y = 1, z = 0. Тогда каноническое уравнение прямой будет иметь вид
Каноническое уравнение прямой в пространстве имеет вид
где (x[sub]0[/sub], y[sub]0[/sub], z[sub]0[/sub]) - произвольная точка прямой, {l, m, n} - направляющий вектор прямой. Если прямая является пересечением двух плоскостей, уравнения которых известны (как в данном случае), то в качестве направляющего вектора можно взять векторное произведение нормальных векторов этих плоскостей, которое по определению перпендикулярно обоим векторам, следовательно, параллельно плоскостям, а значит, и прямой. В качестве точки можно взять любую, чьи координаты удовлетворяют уравнениям обеих плоскостей.
Координаты нормальных векторов определим непосредственно из уравнений соответствующих плоскостей:
Их векторное произведение будет равно
то есть направляющий вектор прямой {-1, 17, -6}. Координаты точки получаем, решая систему
Её решением будет, например, x = -1, y = 1, z = 0. Тогда каноническое уравнение прямой будет иметь вид
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